The complexity of finding minimum-length generator sequences

Abstract

The computational complexity of the following problem is investigated: Given a permutation group specified as a set of generators, and a single target permutation which is a member of the group, what is the shortest expression for the target permutation in terms of the generators? The general problem is demonstrated to be PSppace-complete and, indeed,is shown to remain so even when the generator set is restricted to contain only two permutations. The restriction on generator set cardinality is the best possible, as the problem becomes soluble in polynomial time if the generator set cantains only one permutation. An interesting feature of this problem is that it does not fall under the headings of ‘two person games’ or ‘formal languages’ which cover the great majority of known PSpace-complete problems. Some restricted versions of the problem, in which the generator set is fixed rather than being part of the problem instance, are also investigated and shown to be computationally tractable. One result of this kind is that determining the most compact expression of a permutation in terms of ‘cyclicly adjacent transpositions’ can be achieved in polynomial time. Thus, from an initial arrangement of distinct objects on a circle, one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement. Surprisingly, this problem appears substantially more difficult to solve than the related one (for which a solution has been known for some time) in which the objects are arranged on a line segment.